Double expectation (Not law of iterated expectation) Suppose I want to calculate 
$$E_X[E_Y{g(X,Y)}]$$
Am I supposed to calculate
$$ \int \int g(x,y) f(x) f(y) \,dx\,dy$$ 
or 
$$\int \int g(x,y) f(x,y) \,dx\,dy $$
where $f(x)$ is marginal pdf and $f(x,y)$ is joint. I assume expectation exists?
 A: I wonder if any respectable probabilist ever wrote $\operatorname E_Y[g(X,Y)].$ In standard usage, the random variables $X,Y$ have some joint distribution, and $\operatorname E[g(X,Y)]$ means an integral with respect to that joint distribution. If you want to integrate with respect to $Y,$ with $X$ fixed, then you're finding a conditional expected value, and it's written as $\operatorname E[g(X,Y)\mid X].$ That is a quantity whose value is determined by the value of $X.$ It is random because $X$ is random. Since it is random, one may ask what its expected value is, and that is denoted by $\operatorname E[ \operatorname E(g(X,Y)\mid X]],$ and that is an iterated integral whose value is the same as that of $\operatorname E[g(X,Y)].$ The reason for computing it as an iterated integral in that way is often simply that you have a method for calculating each of the two integrals.
One should not use the same symbol, $f,$ two refer to three different functions. What is $f(x)$ when $x=3$ and what is $f(y)$ when $y=3$? They're both $f(3).$ What does that mean if the two $f$s are two different functions? One can write $f_X(3)$ and $f_Y(3)$ and know what it means (with capital $X$ and capital $Y,$ meaning the two random variables).
Now we have
$$
\operatorname E[g(X,Y)] = \iint g(u,v)f_{X,Y}(u,v)\, d(u,v)
$$
or synonymously
$$
\operatorname E[g(X,Y)] = \iint g(s,t)f_{X,Y}(s,t)\, d(s,t)
$$
etc. Changing the names of the bound variables  doesn't change the value of the integral; one can equally validly call them $x,y.$
A conditional expectation can be written as
$$
\operatorname E[g(X,Y)\mid X=x] = \int g(x,y) f_{Y\,\mid\,X\,=\,x}(v)\, dv.
$$
This is a function of (lower-case) $x.$ That same function evaluated at (capital) $X$ is a random variable, denoted by $\operatorname E[g(X,Y)\mid X].$
A: The subscript in expectation operator generally denotes which density to use; apparently in your notation, $E_Y[.]$ means the expected value of the inner term using the density of random variable $Y$. So, your first answer is correct. The second one is actually the law of iterated expectations, since $f(x,y)=f(x)f(y|x)$, which results in inner expectation of the form $E_{Y|X}$.
